Pareto efficient allocation and core allocation in 2 person exchange economy with discontinuous utility

Question

I understand that with continuous and strictly increasing utility function we can find Pareto efficient allocations by looking at the allocation that satisfies $MRS_1 = MRS_2$. However, I was not sure how we can find Pareto Efficient Allocations for discontinuous utilities (or one continuous and one discontinuous utility).

For instance, given exchange economy with one unit of initial endowment of the good $x$ and $y$, if we have person A's utility function as $$u_A(x,y) = x^\alpha y^{1-\alpha}$$ and person B's preference is represented by lexicographic: given $(x,y)$ and $(x',y')$, B prefers the former if $$x>x'$$ or $$x = x', y>y'$$ In this case, we can't calculate $MRS_B$. Does anyone have any suggestion how can we find the Pareto Efficient Allocation and Core Allocation? Thank you very much for your help in advance!

Answer 1

Here are two pertinent observations:

1. If A has a positive amount of the $x$-good, and $B$ a positive amount of the $y$-good, both can be made strictly better off (why?).

2. There is no efficient allocation in which $A$ has no amount of the $x$-good, but a positive amount of the $y$-good (why?).

Remaining are the allocations in which $B$ gets everything and allocations in which A has a positive amount of the $x$-good and all of the $y$-good. Verify that these are indeed Pareto efficient allocations.

Finding core allocations is not much different. There is a small difference if you define blocking so that all parties involved might be strictly better off. In that case, you can have allocations in which $A$ has none of the $x$-good but some of the $y$-good.